Question

simplify the following algebraic expression
m² + 6m + 5/m² — m — 2​

Answer 1

$\bold\red{\fbox{\sf{Solution}}}$

Adding a fraction to a whole

Rewrite the whole as a fraction using m2 as the denominator :

m2 + 6m (m2 + 6m) • m2

m2 + 6m = ——————— = ——————————————

1 m2

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Pull out like factors :

m2 + 6m = m • (m + 6)

Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

m • (m+6) • m2 + 5 m4 + 6m3 + 5

—————————————————— = ————————————

m2 m2

Subtracting a whole from a fraction

Rewrite the whole as a fraction using m2 as the denominator :

m m • m2

m = — = ——————

1 m2

Find roots (zeroes) of : F(m) = m4 + 6m3 + 5

Polynomial Roots Calculator is a set of methods aimed at finding values of m for which F(m)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers m which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is 5.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,5

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 0.00 m + 1

-5 1 -5.00 -120.00

1 1 1.00 12.00

5 1 5.00 1380.00

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that

m4 + 6m3 + 5

can be divided with m + 1

Polynomial Long Division

Dividing : m4 + 6m3 + 5

("Dividend")

By : m + 1 ("Divisor")

dividend m4 + 6m3 + 5

- divisor * m3 m4 + m3

remainder 5m3 + 5

- divisor * 5m2 5m3 + 5m2

remainder - 5m2 + 5

- divisor * -5m1 - 5m2 - 5m

remainder 5m + 5

- divisor * 5m0 5m + 5

remainder 0

Quotient : m3+5m2-5m+5 Remainder: 0

Find roots (zeroes) of : F(m) = m3+5m2-5m+5

See theory in step 4.2

In this case, the Leading Coefficient is 1 and the Trailing Constant is 5.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,5

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 14.00

-5 1 -5.00 30.00

1 1 1.00 6.00

5 1 5.00 230.00

Polynomial Roots Calculator found no rational roots

Adding up the two equivalent fractions

(m3+5m2-5m+5) • (m+1) - (m • m2) m4 + 5m3 + 5

———————————————————————————————— = ————————————

m2 m2

Factoring: m4 + 5m3 - 2m2 + 5

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: -2m2 + 5

Group 2: m4 + 5m3

Pull out from each group separately :

Group 1: (-2m2 + 5) • (1) = (2m2 - 5) • (-1)

Group 2: (m + 5) • (m3)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Find roots (zeroes) of : F(m) = m4 + 5m3 - 2m2 + 5

See theory in step 4.2

In this case, the Leading Coefficient is 1 and the Trailing Constant is 5.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,5

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 -1.00

-5 1 -5.00 -45.00

1 1 1.00 9.00

5 1 5.00 1205.00

Polynomial Roots Calculator found no rational roots

Answer : m4 + 5m3 - 2m2 + 5

——————————————————

m2

Answer 2

$\huge \bf \pink {answer}$

Adding a fraction to a whole

Rewrite the whole as a fraction using m2 as the denominator :

m2 + 6m (m2 + 6m) • m2

m2 + 6m = ——————— = ——————————————

1 m2

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Pull out like factors :

m2 + 6m = m • (m + 6)

Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

m • (m+6) • m2 + 5 m4 + 6m3 + 5

—————————————————— = ————————————

m2 m2

Subtracting a whole from a fraction

Rewrite the whole as a fraction using m2 as the denominator :

m m • m2

m = — = ——————

1 m2

Find roots (zeroes) of : F(m) = m4 + 6m3 + 5

Polynomial Roots Calculator is a set of methods aimed at finding values of m for which F(m)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers m which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is 5.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,5

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 0.00 m + 1

-5 1 -5.00 -120.00

1 1 1.00 12.00

5 1 5.00 1380.00

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that

m4 + 6m3 + 5

can be divided with m + 1

Polynomial Long Division

Dividing : m4 + 6m3 + 5

("Dividend")

By : m + 1 ("Divisor")

dividend m4 + 6m3 + 5

- divisor * m3 m4 + m3

remainder 5m3 + 5

- divisor * 5m2 5m3 + 5m2

remainder - 5m2 + 5

- divisor * -5m1 - 5m2 - 5m

remainder 5m + 5

- divisor * 5m0 5m + 5

remainder 0

Quotient : m3+5m2-5m+5 Remainder: 0

Find roots (zeroes) of : F(m) = m3+5m2-5m+5

See theory in step 4.2

In this case, the Leading Coefficient is 1 and the Trailing Constant is 5.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,5

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 14.00

-5 1 -5.00 30.00

1 1 1.00 6.00

5 1 5.00 230.00

Polynomial Roots Calculator found no rational roots

Adding up the two equivalent fractions

(m3+5m2-5m+5) • (m+1) - (m • m2) m4 + 5m3 + 5

———————————————————————————————— = ————————————

m2 m2

Factoring: m4 + 5m3 - 2m2 + 5

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: -2m2 + 5

Group 2: m4 + 5m3

Pull out from each group separately :

Group 1: (-2m2 + 5) • (1) = (2m2 - 5) • (-1)

Group 2: (m + 5) • (m3)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Find roots (zeroes) of : F(m) = m4 + 5m3 - 2m2 + 5

See theory in step 4.2

In this case, the Leading Coefficient is 1 and the Trailing Constant is 5.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,5

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 -1.00

-5 1 -5.00 -45.00

1 1 1.00 9.00

5 1 5.00 1205.00

Polynomial Roots Calculator found no rational roots

Answer : m4 + 5m3 - 2m2 + 5

——————————————————

m2